The Twisted Enneper Surface

<< Own`Mesh`

Weierstrass Data

Weierstrass[{G_, η_}, z_]:=Re[∫(({{-((-1 + (G^2))/(2 G))}, {((I (1 + (G^2)))/(2 G))}, {1}}))η \[DifferentialD]z]

G[z_] := P[z]/Q[z];
η[z_] := P[z] Q[z]/z^k;

P[
          z_] := (((ζ^2) z ((z^2) - (ζ^2)))/(-1 - (ζ^4)));
Q[z_] := (z^2) - (1/(ζ^2));
k = 4;

We check that there are no (vertical) periods around the punctures:

Weierstrass[{G[z], η[z]}, z] // Simplify

f[
          z_] := {-(1/6) Re[(((z^6) (ζ^8) + (((1 + (ζ^4)))^2) - 3 (z^4) (ζ^4) (1 + 2 (ζ^4) + 2 (ζ^6) + (ζ^8)) - 3 (z^2) (ζ^2) (2 + 4 (ζ^4) + 2 (ζ^8) + (ζ^10)))/((z^3) (ζ^4) (((1 + (ζ^4)))^2)))], -(1/6) Im[(((z^6) (ζ^8) - (((1 + (ζ^4)))^2) + 3 (z^4) (ζ^4) (1 + 2 (ζ^4) - 2 (ζ^6) + (ζ^8)) + 3 (z^2) (ζ^2) (2 + 4 (ζ^4) + 2 (ζ^8) - (ζ^10)))/((z^3) (ζ^4) (((1 + (ζ^4)))^2)))], -(1/2) Re[(((-1 + (z^4)) (ζ^2) - 2 (z^2) (1 + (ζ^4)) Log[z])/((z^2) (1 + (ζ^4))))]};

n[w_] := StereographicProjection[G[w]]

nx = 15;
ny = 20;

r0 = 8.5;

domain = 
        MeshApply[Exp, 
          RectangularDomain[NRange[-Log[r0], Log[r0], nx], 
            NRange[0, 2  Pi, ny]]];

Choose Dehn Twist Parameter

ζ = 3.  E^(Pi/4  I);

fr1 = 
        MeshPlot3D[{f[x + I y], n[x + I y]}, {x, y}, domain];

Show[gr = Mesh3DToGraphics3D[fr1]];